We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space X. Given a bounded linear operator T defined on X and a discrete almost automorphic function f(n), we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Delta u(n) = Tu(n) + f(n). We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation Delta u(n) = Tu(n) + g(n, u(n)) assuming that g(n, x) is discrete almost automorphic in n for each x is an element of X, satisfies a global Lipschitz type condition, and takes values on X. Copyright (C) 2009 Daniela Araya et al.